Reflect, then slide by half of something

A glide reflects across a plane and then slides along it. The plane’s job is done by the reflection; the slide is the part that has to be justified, and justifying it enumerates the whole family.

The constraint #

Apply a glide twice. The reflections cancel — reflecting across the same plane twice is the identity — and the two slides add. So the square of a glide is a pure translation by twice its slide, and that translation has to be a lattice vector, because a symmetry with no rotation part is a lattice translation.

The slide is therefore half of a lattice vector, lying in the plane.

That is the whole constraint and it is worth noticing how much it rules out. The slide cannot be a third of anything, or a quarter, or an arbitrary fraction — the doubling condition admits halves and nothing else. It cannot point out of the plane, because a glide reflects across the plane and a component perpendicular to it would be removed by moving the origin, which is exactly what “the plane can be put anywhere along its own normal” means.

Three candidates, and then a fourth #

Take a plane spanned by two cell vectors u and v. Which lattice vectors lie in it? Every integer combination of u and v, so the halves available are u/2, v/2 and (u + v)/2 — three, and no more, because (u − v)/2 differs from (u + v)/2 by the lattice vector v and is the same operation.

Three slides, three glides. The first two are the axial glides, named for whichever cell vector they slide along: a, b or c depending on the plane. The third is the diagonal glide, written n, sliding along half a face diagonal.

Now centre the lattice. In an F-centred cell, (u + v)/2 is itself a lattice vector — that is what face centring means. So the doubling condition, which asks that twice the slide be a lattice vector, is satisfied by half of it: (u + v)/4.

That is the d glide, the diamond glide, and the enumeration has just explained the thing about it that is usually stated as a rule. It appears only in F- and I-centred groups because in a primitive lattice its square is not a lattice translation, so it is not a symmetry of anything periodic.

The fifth letter, and where it lives #

The d glide is rare and its structures are famous. It is the glide of diamond itself — space group Fd3̅m, where the two interpenetrating face-centred carbon sublattices are related by a quarter-diagonal glide — and of the spinels, and of silicon and germanium, which have diamond’s structure.

That is worth dwelling on, because it is the clearest case in this field of two independent routes reaching the same fact. The enumeration says the d glide slides by a quarter because centring makes the half a lattice vector. The diffraction calculation, which knows nothing about glides and only sums exponentials over the group’s operations, produces a condition modulo four. Neither derivation mentions the other and they agree.

Why there are three axial letters and not two #

A small point of bookkeeping that is worth getting right, because the numbers do not quite say what they look like.

Any given plane contains exactly two cell directions, so it offers exactly two axial glides. A plane perpendicular to c contains a and b, so it can be an a glide or a b glide and cannot be a c glide — a c glide perpendicular to c would slide out of its own plane, which is not a glide at all.

So the three letters a, b and c are not three glides available in one place. They are one mechanism — slide by half a cell edge lying in the plane — named by which edge, and which two of the three letters are available depends on which way the plane faces. Counting them as three is a count of symbols.

That matters for reading a symbol. In Pbcm the three letters are in the three orthorhombic positions, so b is a plane perpendicular to a sliding along b, c is a plane perpendicular to b sliding along c, and m is a plain mirror perpendicular to c. Each letter has to be legal for its position, and a symbol like Paba is not a symbol at all because the first position cannot carry an a.

The plane already had one of these #

The plane has a glide, and this site has an essay about it whose finding transfers exactly.

In two dimensions the plane’s glide slides by half a cell edge, and there is only one kind, because a line has only one direction to slide along. The rectangular lattice offers u/2 and nothing else; the diagonal case does not arise because the “plane” is a line and it contains one lattice direction.

What the cm essay found was subtler and it is worth repeating here in three dimensions. cm’s glide is a fact about the cell, not about the pattern. Described on the conventional centred rectangular cell, cm has four operations and one of them is a glide. Described on the primitive rhombic cell of the same lattice, it has two operations and neither is a glide, because the centring translation is a lattice vector in that basis and the “glide” is the mirror composed with it.

The same thing happens in space, wherever a centring translation lies in a mirror plane. A mirror in a C-centred group, composed with the (½, ½, 0) centring, gives a glide — and whether that counts as a distinct operation depends on which cell is being used. The round trip here treats centring translations as operations of the group, so the answer is always given with respect to the conventional cell, and the conventional cell is stated.

Where the notation puts the letter #

In a Hermann–Mauguin symbol a glide’s letter names the direction it slides, and the position in the symbol names the direction the plane faces. Those are two different directions and confusing them is the single commonest mistake in reading these symbols.

Pna2₁ has three positions after the lattice letter, one per axis of an orthorhombic cell. The first is n: perpendicular to a, a plane that slides along a face diagonal. The second is a: perpendicular to b, a plane sliding along the a direction. The third is the polar axis, along c, carrying a 2₁ screw.

So the letter a in that symbol is in the second position and does not refer to the a axis as a plane normal — it refers to the a axis as a slide direction, in a plane that faces b. The symbol reads position-first and letter-second, and reading it letter-first gives a plausible and wrong answer.

Composition, and the glide nobody chose #

The other way to get a glide is to not ask for one.

A rotation composed with a mirror gives a reflection; a rotation composed with a glide gives another glide, or sometimes a mirror. And a mirror composed with a translation lying in its plane gives a glide directly, with no rotation involved — which is exactly what a centring vector does when it lies in a mirror plane.

That is the cm mechanism and it is common in space. Any C-centred group with a mirror perpendicular to c acquires a glide, because the centring vector (½, ½, 0) lies in that plane, and the mirror composed with it slides by half a face diagonal — an n glide. Nobody put it there.

The consequence for the machinery here is that a group is never described by a list of its operations written down in advance. The generators are stated, the closure is run, and whatever the closure produces is what the group has. A list of operations typed out by hand would be missing precisely the ones that are hardest to anticipate, which is the subject of the next rung.

What a glide costs a structure #

Like a screw, a glide has no fixed point — a point on the plane reflects to itself and then slides, so it moves. That means no atom sits on a glide plane, and a glide contributes nothing to the list of special positions.

The consequence is the same arithmetic the screw essay reached. A group whose only non-translational operations are glides acts freely, and every orbit has the full order of the group; the number of formula units in the cell is a multiple of that order, with no exceptions available.

A mirror, by contrast, is generous. Every point of a mirror plane is fixed by it, so a mirror gives a two-dimensional sheet of special positions, and a molecule with its own mirror symmetry can sit on one and halve its contribution to the cell. Mirrors are the most abundant source of special positions in any group that has them, which the Wyckoff analysis found in the plane too: sixty of a hundred and forty-four sample points in p4m lie on a mirror, against two at the fourfold centres, because a line has room and a point does not.

So the difference between Pmm2 and Pcc2 is not decorative. One offers two perpendicular sheets of special positions and the other offers none, and a molecule choosing between them is choosing whether it can economise.

Two glides at right angles make a screw #

The last thing worth showing is what glides do to each other, because it is the mechanism behind four of the ten groups in the mm2 enumeration and it is not obvious.

Compose two glides whose planes are perpendicular. The two reflections compose to a two-fold rotation about the line where the planes meet — that part is the same as for two mirrors. The two slides add, and the component along the intersection line survives, because both slides lie in their own planes and the intersection lies in both.

So if either glide slides along the line where the planes meet, the product is a two-fold screw about that line.

The arithmetic is worth doing once because it explains the last character of four space-group symbols. If the first glide slides c₁ along the intersection and the second slides c₂, the product slides c₁ + c₂ — so the product is a screw exactly when c₁ + c₂ is a half, which is exactly when one of the two is a half and the other is zero. Both zero gives a rotation; both halves gives a whole cell, which is a lattice translation, so again a rotation.

That is a small parity argument and it accounts for a quarter of an entire arithmetic class.

Where the enumeration stops #

Three limits.

The five letters are a naming convention over three geometric cases. There are not five kinds of glide; there are axial glides, of which there are up to three per plane depending on which cell vectors lie in it, one diagonal glide, and one diamond glide available only under centring. The letters a, b and c are the axial case named by direction, so the count of five is a count of symbols rather than of mechanisms. That is worth saying because “five glides” and “eleven screws” sound like comparable counts and are not: the eleven really are eleven distinct operations.

The d glide’s availability depends on the centring, not on the system. It appears in F- and I-centred groups, and the enumeration gives the reason — the half-diagonal has to be a lattice vector before a quarter of it can be a glide. There is no d glide in any primitive group, and the machinery here would refuse one: its square would not be a lattice translation and the closure would run away.

Nothing here says which glides a real crystal has. A glide is available if the group has it, and whether a structure adopts a group with one is a question about packing. Glide planes are common in molecular crystals because they let a molecule pack against a shifted copy of itself rather than a directly-facing one, which for most molecular shapes is denser — but that is an argument about shapes, not about symmetry, and this machinery has no access to it.

Who named them, and why the letters are what they are #

The letters are Hermann’s and Mauguin’s, from the early 1930s, and they were chosen to be read rather than looked up: a, b and c name the axis the plane slides along, so the symbol contains the geometry.

n is for net, or diagonal depending on which account is being read, and it is the only letter of the five that is not the name of an axis. d is for diamond, after the structure, which is unusual — a symmetry operation named for a substance rather than for what it does. The reason is historical rather than principled: the operation was recognised in diamond’s structure before it was recognised as a general possibility, and by the time the enumeration made clear that it was one of a family the name had stuck.

There is a sixth letter in modern editions of the Tables, e, introduced in 1992 for a double glide — a plane that is simultaneously an a glide and a b glide, which happens when a centring translation makes both slides available on the same plane. It is not a new mechanism and it does not extend the enumeration above; it is a notational device for a case the old symbols described inconsistently, and five different space-group symbols were changed to use it. This site’s enumeration produces the five mechanisms and does not model the e convention, which is a choice about notation rather than about symmetry.

What both this rung and the last one keep running into is composition producing operations nobody chose. That is the next essay, and it is the reason a group is never given here as a list.

Three, because a rank-two lattice has four halves #

The enumeration finds three candidates by trying them, and there is a one-line reason the number is three rather than some other small number — worth having, because it is the reason the answer does not change with the crystal system.

The slides available in a plane are the half-lattice vectors it contains, counted modulo the lattice. Write $L$ for the lattice vectors lying in the plane, a rank-two lattice. A glide’s slide must be half of one of them, and two slides differing by a whole lattice vector give the same operation, since a symmetry composed with a lattice translation is another symmetry of the same kind. So the possible slides are the elements of $\tfrac12 L$ modulo $L$.

That quotient has exactly four elements. Halving each of two independent generators and taking all combinations gives nothing, half of one, half of the other, and half of their sum — the group of two binary digits, whatever the shape of the cell.

One of the four is the mirror. The zero slide is the operation that slides by nothing, so it is not a glide at all. Three remain, and they are the two axial glides and the diagonal glide, in every plane of every crystal system.

Centring is the only thing that changes the answer, because centring changes what $L$ is. If the half-diagonal is itself a lattice vector then $L$ is larger, its halves are quarters of the conventional diagonal, and a fourth kind of glide appears in the same quotient. The arithmetic is unchanged; the lattice it is performed on is not.

The letter the Tables added #

There is a sixth letter in current use, and its history is a good illustration of how a symbol can be right and still mislead.

Some planes carry two glides at once. In a lattice centred on the face containing the plane, a plane can be simultaneously an $a$-glide and a $b$-glide — the same plane, two different slides, both genuine symmetries of the crystal, because the centring translation carries one onto the other. Nothing in the enumeration forbids it; the plane simply has more than one of the four elements above realised on it.

The old symbols named one and hid the other. A group whose plane was both was written with whichever letter convention preferred, so two groups differing in a real way could look alike, and one group could be written two ways by two people following the same rules.

The letter e names the pair. Introduced by the International Union of Crystallography in 1992 and standard in the Tables since, e means a double glide plane — both axial slides present. Five space-group symbols changed as a result: Abm2 became Aem2, Aba2 became Aea2, Cmma became Cmme, Ccca became Ccce, and Cmca became Cmce.

None of the groups changed. No symmetry was discovered and no enumeration was corrected; the symbols were made to say what the groups had always contained. That is the useful thing about the episode — a naming convention that reports one of several equally valid features is not a description, and the fix was to stop choosing.

And the older symbols are still everywhere. A structure published before the change carries the old letter, and databases hold both, so a reader comparing two entries has to know that Cmca and Cmce are the same group. That is the practical cost of the correction, and it is the ordinary cost of any correction to a notation that has been in use for a century.